A stochastic approach for spectrum sensing and sensor selection in dynamic cognitive radio sensor networks

A stochastic approach for spectrum sensing and sensor selection in dynamic cognitive radio sensor networks

Physical Communication 37 (2019) 100879 Contents lists available at ScienceDirect Physical Communication journal homepage: www.elsevier.com/locate/p...

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Physical Communication 37 (2019) 100879

Contents lists available at ScienceDirect

Physical Communication journal homepage: www.elsevier.com/locate/phycom

Full length article

A stochastic approach for spectrum sensing and sensor selection in dynamic cognitive radio sensor networks ∗



Karel Toledo a , , Hector Kaschel a , , Jorge Torres b , a b



Department of Electrical Engineering, University of Santiago de Chile, Santiago, GA, 9160000, Chile Department of Signal Theory and Communications, Carlos III University of Madrid, Leganes, Madrid, GA, 28911, Spain

article

info

Article history: Received 25 April 2019 Received in revised form 9 August 2019 Accepted 29 September 2019 Available online 3 October 2019 Keywords: Energy efficiency Spectrum sensing Dynamic CRSN Markov chains Stochastic programming

a b s t r a c t Cognitive Radio Sensor Networks (CRSN) is currently demanding to deal with spectrum scarcity through opportunistic spectrum access solutions. Opportunistic access to the available spectrum is achieved by determining spectrum holes, which in turn demands to run further signal processing operations on network nodes. Consequently, energy consumption to support these processing algorithms is increased and thus, it remains a major concern in CRSN. Some solutions address this issue via sensor selection during spectrum sensing in static CRSN. In this case, certain nodes participate in cooperative spectrum sensing (CSS) to guarantee proper performance, and the remaining nodes go to sleep to extend network battery lifetime. However, this strategy becomes difficult to apply to mobile sensor networks, where nodes change positions dynamically. This is the case of mobile nodes on Cognitive Radio Internet of Things (CR-IoT) networks, where sensor nodes consume significant energy to support CR operations. Due to the random displacement of nodes, new solutions must be developed in contrast to previous strategies based on on-off nodes from static CRSN, this to cooperate between nodes and also to reduce consumed energy. This paper reports a novel energyefficient sensor selection technique applicable to dynamic CRSN. Stochastic approaches are developed to describe and determine the minimum total number of awake sensors to participate in CSS. This approach is particularly suited for solutions in which conditions, such as the position of nodes, change randomly. Proposed solution is obtained through the use of ‘‘here-and-now" approach considering statistic features of the random distance between each sensor node and a given fusion center. The methodology achieves energy consumption levels comparable to ‘‘wait-and-see" approach for networks of reduced size. Additionally, we provide further insights into the statistical nature of the problem to state a proper problem formulation, then to devise solutions accordingly. The analysis and performance of the proposed solution are discussed and illustrated with the aid of simulations. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Cognitive Radio Sensor Networks (CRSN) represent a recent growing technology with a variety of applications such as smart homes and buildings, intelligent transportation, health care, military, industrial processes, precision agriculture, security and environment [1]. A CRSN is comprised of a large number of small devices to support CR capabilities, namely secondary users (SUs). On the other hand, primary users (PUs) are devices with legacy rights on spectrum usage. SUs demand for available spectrum holes to increase transmission opportunities, those detected and transmitted to a fusion center (FC), where data are merged to have a final decision regarding spectrum availability. To successfully perform cooperative spectrum sensing (CSS), fusion center ∗ Corresponding authors. E-mail addresses: [email protected] (K. Toledo), [email protected] (H. Kaschel), [email protected] (J. Torres). https://doi.org/10.1016/j.phycom.2019.100879 1874-4907/© 2019 Elsevier B.V. All rights reserved.

can merge sensing results using two main rules: data fusion (soft rule) or decision fusion (hard rule), as discussed in [2]. Additionally, SUs are powered by a battery which supplies energy to one or more detection units, processors, memory blocks and wireless transmission and reception modules. One of the major challenges on CRSN applications is to minimize energy consumption of SUs aiming to extend battery lifetime. Some authors address this issue through sensor selection during spectrum sensing phase in static CRSN [3–10]. Certain nodes or SUs are awake to participate in cooperative spectrum sensing while otherwise they are on sleep mode to reduce energy consumption. These reported solutions determine the minimum number of nodes awake while satisfying detection performance simultaneously. However, CRSN has to deal with the new paradigm of Internet of Things (IoT). For instance, the convergence between 5G and IoT communications has been addressed in [11], where a

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K. Toledo, H. Kaschel and J. Torres / Physical Communication 37 (2019) 100879

wireless network comprised by sensor nodes with CR capabilities performs dynamics spectrum sharing and the power optimization unit is deployed to improve the system throughput. Another solution focused on maximizing the system throughput, subject to constraints on energy requirement and interference, merges the wireless power transfer and wireless information transfer technologies in one 5G-based system to guarantee both throughput and energy efficiency, simultaneously [12]. In addition, displacement of nodes in IoT is an essential issue [13,14] not currently analyzed. These concerns motivate to devise energy efficient sensor selection methods to report applications on dynamic networks. Mobility of networks introduces variability on position of SUs, then optimal selection of awake nodes for CSS must be described by dynamic rules. In this scenario, current static solutions in [3–10] are limited to address proper performance of dynamic networks due to assumption of constant distance from each SU to FC and PU. Displacement of nodes changes network architecture dynamically which in turn establishes limited application of static reported solutions regarding sensor selection for CSS. In this regard, stochastic programming provides the means to deal with dynamic environments. For instance, ‘‘wait-and-see" [15] and ‘‘here-and-know’’ [16] are two given approaches for modeling optimization problems which embody a random variable formulation -based. These approaches are able to deal with random parameters which in turn offer the possibility of handling random positions of moving nodes. Current work is inspired by works presented in [4,6] for static CRSN and further analyzed in [7,10]. However, we develop a stochastic model for energy efficient sensor selection. To the best of our knowledge, stochastic programming had not been previously applied to this problem. Main contributions of this paper are listed below.

• We model a CRSN composed of sensor nodes performing random movement based on Markov chains which are suitable for Internet of Mobile Things environments [17,18]. Then, we derive a new sensor selection strategy for dynamic networks to minimize energy consumption on spectrum sensing operations based on stochastic optimization approaches. An approximate solution to this problem stems from application of ‘‘here-and-now" approach [16], which analyzes random movement of SUs. Two different criteria are presented considering statistic features of distance between each sensor node and FC: Expected Value and Expected Value Standard Deviation. • An iterative algorithm is proposed to select the proper awake sensor nodes for spectrum sensing operations, while remaining nodes are placed in sleep mode to extend network battery lifetime. This algorithm avoids the repetition of a static solution on each time slot in comparison with ‘‘waitand-see’’ approach [19]. Also, computational complexity is improved with respect to the exhaustive search algorithm. • We derive asymptotic behavior of network for different case scenarios such as different network sizes, position of FC and PU, and available SUs. This to illustrate the total number of awake nodes and their corresponding consumed energy in spectrum sensing phase. The structure of this paper is as follows. Section 2 presents the system model and main assumptions on spectrum sensing and random movement model. Formulation of the stochastic problem is introduced in Section 3. Section 4 describes the two proposed stochastic criteria. Analysis of implemented algorithm is discussed in Section 5. Section 6 presents illustrative examples followed by concluding remarks and envisaged future work exposed in Section 7.

Fig. 1. A sample of the initial position of the system model for CRSN.

2. System model We consider a CRSN composed by N SUs with random movement distributed over a rectangular field of side s, a fixed FC is located at the center and there is only one PU as shown in Fig. 1. SUs are able to access licensed spectrum bands from PU avoiding interference. To this purpose, SUs are continuously sensing available spectrum looking for spectrum holes. Collected information from several SUs is sent to the FC to merge and establish a final decision concerning the PU presence on a given frequency band. We assume equal spectrum sensing duration δ for all sensors given by a multiple of 1/fs . Quantity fs represents sampling frequency value of the received signal at the j-th sensor, j ∈ N. Finally, total number of processed samples is given by δ fs . 2.1. Cooperative spectrum sensing Typically, the energy detection method [20] is applied by each SU to establish the decision statistic. Then, the following two hypothesis, H1 and H0 , are defined as: H1 represents busy channel due to the presence of the PU and H0 represents idle channel due to the absent of the PU given by: H1 : yj [n] = hj [n]xj [n] + uj [n],

(1)

H0 : yj [n] = uj [n],

(2)

where n = 1, 2, . . . , δ fs is the time index, hj [n] is the channel impulse response between each SU and the PU, xj [n] is the signal of interest transmitted from the PU and uj [n] is an i.i.d. Gaussian noise with zero mean and variance σu2j . Consequently, based on energy detection principles the decision rule can be stated as: H1 if Ej ≥ ϵ or H0 if Ej < ϵ . Parameter ϵ represents the detection threshold, this value is established as a multiple of noise variance σu2j , and Ej is the energy of received signal at the j-th

∑δ f

2 s sensor defined by Ej = δ1f n=1 |yj [n]| . Additionally, false alarm s and detection probabilities for a given j-th sensor are defined as follows [20]:

) ) √ ϵ Pfj = P(Ej > ϵ|H0 ) = Q −1 δ fs , σu2j (( ) )√ ϵ δ fs Pdj = P(Ej > ϵ|H1 ) = Q − γj − 1 , σu2j 2γj + 1 ((

(3)

(4)

where term γj is the signal-to-noise ratio (SNR) of PU and Q is the complementary distribution function of the standard Gaussian

K. Toledo, H. Kaschel and J. Torres / Physical Communication 37 (2019) 100879

∫∞

3

2 − t2

distribution given by Q (x) = √1 x e dt. Each node sends 2π one bit to FC according to statistical test and a combined decision may be implemented through AND, OR or Majority rules. In this work, OR rule is adopted where the PU is considered to be present if at least one sensor claims the presence of a given PU. Hence, the global probability of false alarm and the global probability of detection can be obtained as follows [20]: PF = 1 −

N ∏

(1 − ρj Pfj ),

(5)

j=1

PD = 1 −

N ∏

(1 − ρj Pdj ),

(6)

j=1

considering ρj ∈ {0, 1}, i.e. ρj = 1 if sensor participates in spectrum sensing, otherwise ρj = 0. Variable ρj in (5) and (6) establishes a relation between performance results and sensor selection strategies.

Fig. 2. Path traveled by one sensor mode in 500 steps.

2.2. Mobility model Mobility models for mobile networks and internet services have been extensively explored and some solutions may be suitable for CRSN. Currently, there are two main displacement models: entity and group mobility models. Entity models are focused on individual movements while group mobility models simulate situations in which node decisions on movement depend upon other nodes positions in the group [21,22]. The mobility pattern implemented in this work is motivated by Random Walk entity mobility model discussed in [23] provided we do not have any assumptions on group mobility models. Movement of nodes is modeled by a stochastic process X (t) fully compliant with the Markovian property. In this regard, we assume time is slotted by t = 1, 2, . . . , T and each SU may perform at most one movement per time slot. This Markov chain presents a finite number of states according to the side of the given rectangular field. Future evolution of the process, once it is in a given state, depends only on the current state and not on past states. Considering X (t) = i represents the current process in state i during time slot t, then we suppose whenever the process is in state i, there is a fixed probability pik to be in next state k as follows [24]: P {X (t + 1) = k|X (t) = i} = pik ,

(7)

∑s

where pik > 0 and k=0 pik = 1, i = 0, 1, 2, . . . , s. Let P denotes the general one-step transition probabilities square matrix of order (s + 1), terms q, r , p represent the probabilities to move to the left, remain in current position or go to the right, respectively, then for a given node:



[

P = pik

]

r0 ⎢q1 ⎢ ⎢0

⎢. =⎢ ⎢ .. ⎢ ⎢0 ⎣0 0

p0 r1 q2

.. .

0 0 0

0 p1 r2

.. .

qi 0 0

.. .

... ... ... .. .

ri

pi

qs−1 0

rs−1 qs

0 0 p2



0 0 ⎥ ⎥ 0 ⎥

.. ⎥ ⎥ . ⎥, ⎥ 0 ⎥ p ⎦

two-dimensional area, we need to simulate this two stochastic processes X (t) and Y (t). Based on these rules, SUs can reach one of nine potential positions on each time slot t, as illustrated in Fig. 1 by a white circle, for instance. That is, to have next displacement following arrow directions or to remain on current position. This movement of SUs has a direct impact on energy consumption value owing to the varying distance to FC. Fig. 2 shows an example of the random movement of the j-th sensor for a network side s = 100 m. FC is placed at coordinate (50, 50), start and end labels represent the starting and the ending points of the path covered by that sensor. In this example, the node finishes its path closer to FC which implies that consumed energy value at end position is smaller than start position. 3. Problem formulation Problem formulation is based on the challenger exposed for CRSN considering the random movement of sensor nodes. This to reduce consumed energy and simultaneously have proper detection performance of spectrum holes. Current work seeks to minimize consumed energy by mobile SUs during spectrum sensing operations subject to constraints on detection performance. In order to describe energy consumption on the network, we need to define two main quantities. The first one, given by Esj , represents the additive contribution of the total amount of consumed energy during sensing operations by each sensor and also, the energy required to perform local decisions. The second one, given by Etj , indicates the total amount of consumed energy by each SU to report sensing operation results. Based on aforementioned definitions, total energy consumption, denoted by ET , is given by [4]:

(8)

s−1

rs

where r0 + p0 = 1, qs + rs = 1, qi + ri + pi = 1 and i = 0, 1, 2, . . . , s. Matrix P in (8) for x and y axes will be used to derive the statistics of random displacement of nodes. Displacement on the vertical axis is modeled by a similar stochastic process Y (t) that satisfies the Markovian property. This random walk has the same properties of the stochastic process X (t) and the matrix of transition probabilities is also derived similar to (8). In order to have next position of each SU on a given

N ∑

ρj (Esj + Etj ),

(9)

Etj = Et-elec + eamp d2j .

(10)

ET =

j=1

where:

Based on (10), parameter Et-elec stands for energy dissipated to run the radio electronics, eamp is the required power amplification and dj is the distance between the j-th SU and the FC. Energy consumed in transmission phase Etj is directly related to the distance dj between each SU and FC [25]. The shorter the value of dj , then the less energy is spent by the j-th sensor as described in (10). In a similar way to the global probabilities in (5) and (6), parameter ρj ∈ {0, 1} has also been included on energy formulation in (9).

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K. Toledo, H. Kaschel and J. Torres / Physical Communication 37 (2019) 100879

Mobile behavior of nodes turns this problem into a stochastic optimization problem. Mobility of SUs introduces variability on node distance dj to FC at each time step, then energy consumption values after substitution of (10) in (9) are also a random time varying quantity. In a comprehensive way, the strategy to extend battery lifetime during spectrum sensing, based on random movement of nodes, can be addressed by the following optimization problem: min E˜ T (ρj , d˜ j ) ρj

(11)

s.t. PF ≤ α, PD ≥ β, ρj ∈ {0, 1}, where decision variable ρj represents a binary parameter, distance d˜ j between each SU and FC is considered to be a random variable which in turn implies that objective function given by E˜ T , is also a random variable. Constraints regarding false alarm and detection global probabilities need to satisfy the threshold parameters α ∈ [0, 1] and β ∈ [0, 1], respectively. Feasible solution to this problem is the vector ρ = [ρj ] for which the random variable E˜ T (ρj , d˜ j ) is minimized and simultaneously guaranteeing constraints on detection performance. Static solutions [3–10] are limited to address proper selection of fixed position awake nodes for CSS to have optimal energy consumption values. Those methods assume prior knowledge of distance between SUs and FC, which in turn reduce applicability on dynamic scenarios. In light of these drawbacks, proposed stochastic optimization problem in (11) deals with random movement of SUs considering dynamic behavior of random parameter d˜ j instead of current reported static solutions mentioned above where dj is a well-known quantity. 4. Proposed solution based on stochastic optimization Stochastic optimization problem given in (11) can be addressed from two viewpoints: ‘‘wait-and-see" and ‘‘here-andnow" approaches. The first one makes a choice about optimal vector ρ after a realization of the random variable d˜ j [15]. On each time slot, this procedure brings the optimal solution based on current positions of nodes, which in turn demands a repetitive execution of a given static solution method on each time instant. As a result, energy consumed in determining values of vector ρ is significantly increased1 [19]. On the other hand, ‘‘here-and-now" approach aims to find solutions prior to the realization of the random variable d˜ j [16]. This approach is based on the stochastic nature of nodes movement behavior instead of its local position at a given particular time instant. In this respect, ‘‘here-and-now’’ approach avoids to repetitive compute solutions on each time slot, then to reduce implementation complexity and then to reduce energy consumption values accordingly. Solutions provided by this method are based on transforming the stochastic problem in (11) into a deterministic equivalent form. In this regard, we look at statistic features of the objective function E˜ T , which in turn is dependent on random parameter d˜ j . Based on this approach, we implement particular solutions defined by expected value and a mixed criterion of expected value and variance of E˜ T . Expected value and variance metrics describe a statistical tendency of energy random variable regarding the mobility model associated with each SU. Based on Chebyshev inequality [24], 1 outputs of E˜ T will be bounded by quantity E[E˜ T ] ± (kV[E˜ T ]) 2 with probability (1 − 1k ). That is, the longer we reduce E[E˜ T ] and V[E˜ T ], the more the values of E˜ T will be decremented accordingly. 1 Further details regarding ‘‘wait-and-see’’ approach for dynamic CRSN have been addressed in a previous conference paper [19].

For instance, considering values of k = 2, 4, 25, then quantity 1 E[E˜ T ] ± (kV[E˜ T ]) 2 will represent a bound to E˜ T with probability 0.5, 0.75 and 0.96, respectively. Based on these principles, we then propose approximate solutions to the problem in (11) by reducing the upper bound of E˜ T . 4.1. Expected value criterion This criterion transforms the stochastic objective function E˜ T in (11) into its expected value. Hence, the solution criterion considers minimizing central tendency of the random variable E˜ T . This solution concept has been frequently used to solve stochastic programming problems [16]. Provided the expected value of a given objective function, this criterion may be applied when probability distribution function of E˜ T is unknown. Moments of Euclidean distance depend on the mobility model discussed in Section 2.2. Under these assumptions, problem presented in (11) can be rewritten by the following deterministic equivalent form: min E E˜ T (ρj , d˜ j )

[

]

(12)

ρj

s.t.

1−

N ∏

(1 − ρj Pfj ) ≤ α

(12a)

(1 − ρj Pdj ) ≥ β

(12b)

j=1

1−

N ∏ j=1

ρj ∈ {0, 1}.

(12c)

Optimal solution to this problem may be obtained by applying an exhaustive search algorithm to find the proper values of ρ ⃗. However, the larger the value of total number of nodes N, the more complex the exhaustive search algorithm becomes with the order of O(N !). Nevertheless, a tractable approach to a static optimization alternative is discussed in [4]. This report proposes to relax the integer problem and a near optimal solution is derived via convex optimization. To simplify, we assume ρj is a continuous parameter so that ρj ∈ [0, 1]. Additionally, we rewrite the global probability of false alarm constraint in (12a) provided values of Pfj defined in (3) have the same values for each SU. This is due to the fact that each SU experiences the same noise variance σu2j and processes equal total number of samples δ fs by using the same threshold ϵ . Upon substituting (3) into (12a) and simplifying we obtain an upper limit to total number of sensor nodes as [26]:

⎢ ⎢ ⎢ ∑ ⎢ ρj ≤ ⎢ ⎣ ( N

ln 1 − Q

j=1

ln(1 − α )

((

ϵ σu2

)√ − 1 δ fs

⎥ ⎥ ⎥ ⎥ ⎥ ) ) ⎦ = M.

(13)

j

Based on (13) and substituting energy consumption equations (9), (10) into (12), the equivalent optimization problem may be formulated as follows: min ρj

s.t.

N ( ∑ [ ]) ρj Esj + Et-elec + eamp E d˜ 2j

(14)

j=1 N ∑

ρj ≤ M

(14a)

j=1

1−

N ∏

(1 − ρj Pdj ) ≥ β

(14b)

j=1

ρj ∈ [0, 1].

(14c)

K. Toledo, H. Kaschel and J. Torres / Physical Communication 37 (2019) 100879

To solve the optimization problem in (14), a Lagrange multiplier is associated to each restriction [27]. The Lagrangian function is expressed as a function of the decision variable ρj as: L(ρj , λ, η) =

N N ( ∑ ∑ [ ]) ρj Esj + Et-elec + eamp E d˜ 2j + η (ρj − M) j=1

(

− λ 1−



) (1 − ρj Pdj ) − β ,

[ ]

(15)

j=1

where λ and η are undetermined Lagrange multipliers. The sign conditions state that λ ≥ 0 and η ≤ 0. A feasible solution must satisfy the first-order necessary condition where partial derivative with respect to the optimization variable is vanishingly small. This imposes that: N ∏ [ ] ∂L (1 −ρk Pdk ) = 0. (16) = Esj + Et-elec + eamp E d˜ 2j +η−λPdj ∂ρj k=1,k̸ =j

The solution to this Lagrange system of equations will be given by nodes for which ρj = 1. Based on relation in (16), there are N equations and N unknown decision variables, for which it is difficult to solve these derived equations for variable ρj . To circumvent this difficulty, there is a reported method that evaluates cost functions for each node to determine values of ρj [4]. These cost functions are ordered from lowest to highest values, then cost functions with lower values will determine preferred nodes to participate in CSS provided these nodes will have lower energy consumption values. Following the procedure in [4], cost functions to solve the system of equations in (16) are given as follows: cost(j) = Esj + Et-elec + eamp E d˜ 2j − λPdj ,

[ ]

(17)

where each element matches those from the condition in (16) except for the last one in which elements from nodes k ̸ = j are not considered. In addition, Lagrange multiplier η is omitted from cost functions provided this parameter value can be assumed to be the same for each SU. Consequently, value of η does not provide any criterion to arrange the given cost functions from lowest to highest values. By using cost functions in (17) a feasible solution to problem in (14) will be given as follows: (a) evaluate cost functions in (17) for each node, (b) activate the smallest total number of nodes with lowest cost function value that will satisfy the global probability of detection in (14b). In addition, the total number of active SUs must not exceed the upper limit M specified in (14a). 4.2. Expected value standard deviation criterion This criterion minimizes expected value and variance of the stochastic objective function E˜ T simultaneously. The goal is not only to reduce the expected value of random consumed energy but also its deviation around the expected value. Similar to expected value criterion, this model is applicable by determining statistical moments of the stochastic objective function E˜ T in (11). The resulting problem formulation by using first and second order -moments of consumed energy E˜ T may be stated as follows: min E E˜ T (ρj , d˜ j ) + kV E˜ T (ρj , d˜ j )

[

]

[

ρj

s.t.

]

root function (·) is monotone and formulation in (18) offers a more tractable model to find optimum solution. In this direction, we may define similar cost functions to (17) according to aforementioned assumptions. In this case, expected value and standard deviation of random variable d˜ 2j are included as shown below: cost(j) = Esj + Et-elec + eamp E d˜ 2j + keamp V d˜ 2j − λPdj .

j=1 N

5

1 2

(18)

[ ]

(19)

Similar to previous criterion, nodes with the smallest cost function and simultaneously satisfying detection constraints are selected to participate in CSS, the remaining nodes go to sleep to extend battery lifetime. Solution based on this criterion ensures the minimum expected value and standard deviation of E˜ T (ρj , d˜ j ), simultaneously. 4.3. Further analysis on solution criteria Cost functions in (17) and (19) offer two solutions to determine which sensors will participate in CSS. Those sensors with smallest cost functions and a fulfilled global probability of detection in (6) given a global probability of false alarm in (5) will provide the approximate solution to stochastic problem in (11). Additionally, cost functions may be further simplified by following some assumptions regarding the statistical moments of nodes movement as well as detection parameters. Simplifications of cost functions in (17) and (19) may be provided in two main directions: by assuming equal movement model and equal probability of detection to each SU. First assumption regarding equal movement model aims to describe a scenario where SUs have equal movement behavior without specifying any particular characteristic of a given node. Second assumption related to equal probability of detection describes rectangular fields of reduced length with applications to pico-cells and femto-cells mobile networks [28]. Both assumptions will establish constant values in case of terms Et-elec , eamp E[d˜ 2j ], keamp V[d˜ 2j ] and λPdj in their respective expressions in (17) and (19). On the other hand, provided that selection of nodes is given by obtaining the lowest cost functions, then constant terms above will not bring any selection criterion. In this case, cost functions may be only evaluated and ordered by using first terms given by Esj in (17) and (19) as follows: cost(j) = Esj .

(20)

However, to have performance metrics we need to determine not only cost function values but also the consumed energy on the network. In this regard, we have to compute statistical moments of energy random variable E˜ T dependent on random distance d˜ 2j in order to establish upper bound values for total energy consumption. Consequently, energy consumption values associated to each criterion may be determined by using appropriate energy objective function as follows: ETm =

N ( ∑ [ ]) ρj Esj + Et-elec + eamp E d˜ 2j ,

(21)

j=1

ETmv =

N ( ∑ [ ] [ ]) ρj Esj + Et-elec + eamp E d˜ 2j + keamp V d˜ 2j ,

(22)

j=1

(14a), (14b), (14c).

Provided we are minimizing the expected value plus k times the variance of random variable E˜ T , then solution obtained by solving (18) seeks to minimize upper bound of random variable E˜ T , which in turn will reduce energy consumption. Term kV[E˜ T ] 1 is equivalent used in (18) instead of (kV[E˜ T ]) 2 provided square

where ETm and ETmv represent consumed energy by expected value and expected value standard deviation criteria, respectively. Based on first assumption, all SUs will follow the same movement rules. In this case, we consider that statistical moments of d˜ 2j have equal values for each node. That is, first and second order -moments of d˜ 2j do not depend on variable j. Hence, first

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K. Toledo, H. Kaschel and J. Torres / Physical Communication 37 (2019) 100879

and second order -moments of random variable of d˜ 2j to be determined on (17) and (19) may be computed as follows:

E d˜ 2j = E d˜ 2 =

[ ]

[ ]

s s ∑ ∑

d2ab Pd2 ,

(23)

ab

a=0 b=0

V[d˜ 2j ] = V[d˜ 2 ] = E[(d˜ 2 − µ)2 ] =

s s ∑ ∑

(d2ab − µ)2 Pd2 , ab

(24)

a=0 b=0

respectively, where parameters a, b are rows and columns indexes of possible locations for SUs on a rectangular lattice of lengths as depicted in Fig. 1, d2ab and Pd2 represent all possible ab square distances from each SU to FC and their probability of occurrence, respectively. Values of Pd2 can be derived based on the equation that ab describes evolution of Markov chains defined by limn→∞ p(n) = limn→∞ p(0)P n through the Chapman–Kolmogorov equation [24]. This by following the mobility model discussed in Section 2.2. Parameter p(0) represents the initial distribution probability and p(n) describes the probability distribution to be in a given state after n transitions. For instance, to describe movement on x-axis, quantity p(n) represents a vector of length (s + 1) to have probabilities of node position in accordance with rectangular field depicted in Fig. 1. Similar description is obtained to describe movement on y-axis. Then, by these two state vector, px (n) and py (n), which describe the probability distribution of movement on x and y axes respectively, the probability of a given node to have distance d2ab to FC will be given by: Pd2 = lim pax (n) · lim pby (n), n→∞

ab

n→∞

a, b ∈ {0, 1, . . . , s},

(25)

where superscript a and b indicate a given a-th and b-th element of vector px (n) and py (n), respectively. On the other hand, provided that ‘‘here-and-now’’ proposed approach computes solutions before the realization of the random variable d˜ j , then next location of nodes cannot be specified in advance. Consequently, probability of detection Pdj will be impossible to determine provided this quantity depends on SNR parameter γj between the given SU and PU, as described in (4). In this regard, we circumvent this concern assuming that our simulation field is a big cluster composed by a circumference containing a rectangular area as displayed in Fig. 1. In this direction, value of Pdj in (4) has equal terms δ fs , σu2j and ϵ to each SU except for SNR parameter γj . This is in accordance with second assumption, where we shall assume that all sensors covering this rectangular area are inside the same cluster and experience the same SNR value. This assumption may be applied when differences between SNR of SUs are less than 1 dB which represents a practical assumption on common scenarios [6]. To accomplish that, distance from PU to cluster center needs to guarantee: Rpu ≥

10 10

0.1

θ

0.1

θ

+1 −1

Rc ,

(26)

where Rc is the cluster radius and θ = 3 is the path loss exponent suggested by Hata model [6]. If this additional requirement is fulfilled, then probability of detecting PU signal Pdj will be equal for each SU. SNR parameter, denoted by γ , is determined only for free space model in terms of the received power PR and noise power N0 as:

γ =

PR N0

=

PT GT GR λ2s (4π d)2 N0

=

)2

PT GT GR Ls N0

,

(27)

where term Ls = 4λπsd establishes the minimum transmission power PT according to frequency band and distance from PU

(

to SU [29]. Antenna gains are denoted by GT and GR , λs = fc c represents the wavelength of the transmitted signal, c is the 8 speed of light 3 · 10 m/s, and fc is the carrier frequency of the transmitted signal. Additionally, CRSN may be categorized as homogeneous or heterogeneous networks whether participating nodes have the same hardware capabilities or different hardware components, respectively. On heterogeneous networks, radio frequency transceivers and processors have different performance between SUs, which in turn implies that consumed energy by each SUs is dependent on specific hardware capabilities. Therefore, sensor nodes must be properly selected to participate in spectrum sensing operations to minimize energy consumption levels. We develop approximate solutions based on cost functions that only differ on energy parameter Esj . This means that SUs with smallest Esj values will have higher priority to be chosen to participate in CSS, the remaining nodes go to sleep mode. On homogeneous networks, we can assume that Es has equal values for each SU. Cost function in (20) depict equal values on each node provided this relation are dependent on a constant parameter. Under this assumption, every SU will have the same priority of being selected to participate in CSS. In this case, final solution to optimization problems just need to guarantee detection performance. Hence, we are only able to determine the total number of SUs to guarantee detection performance. 4.4. Concluding remarks Optimal solutions based on two previous criteria in Sections 4.1 and 4.2 may be derived via exhaustive search algorithm that becomes more complex for larger values of total number of nodes N. Proposed solutions evaluate the same cost function in (20) related to priority of nodes to participate in CSS. The lower the cost function value, then the higher the priority to be selected to participate in spectrum sensing. To find the lowest cost function for each criterion we need to evaluate iteratively expression in (20) for each SU. This may be implemented by an algorithm to perform three different tasks: to evaluate cost functions, ordering the resulting outcomes in ascending order (in case of heterogeneous networks only), then to select the minimum total number of nodes to accomplish with detection performance. Output of this algorithm will bring a suboptimal solution to the stochastic problem in (11). In addition, solutions proposed in Sections 4.1 and 4.2 are only computed once time for the given conditions instead of repeating a static solution for each time slot. That is, based on total number of available nodes, their movement models and specifics energy consumption values to process and transmit information, then we may choose a non-variable appropriate number of awake nodes to guarantee detection performance and reduce energy consumption values. 5. Analysis of the algorithm In this section we describe the iterative algorithm to compute the total number of SUs involved in CSS following the addressed solutions in Sections 4.1 and 4.2. This algorithm is applicable to the two stochastic optimization criteria considering the reduced cost function in (20). A practical solution is formulated in Algorithm 1 considering dynamic behavior of SUs. The algorithm determines the minimum total number of awake SUs to satisfy detection performance, then to extend battery lifetime. In this regard, we choose nodes with minor cost functions to fulfill simplified constraints related to false alarm and detection global probabilities in (14a) and (14b), respectively. Main w hile loop in Algorithm 1, from line 9 to 21, is dependent on parameter M

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which is upper bounded by N. Moreover, a nested loop is executed in line 12 to compute the global probability of detection PD based on selected_nodes array. Hence, computational complexity of proposed algorithm is O(N 2 ) which improves the exhaustive search algorithm of complexity O(N !). Algorithm 1 Stochastic Approaches Inputs: N, s Outputs: ρ , ET 1: Initialize ρ as a 1-by-N array of zeros ˜ 2 based on (23) 2: Compute first and second order moments of d and (24) 3: Compute Pd and M based on (4) and (13) 4: for j = 1 to N do 5: Calculate appropriate cost(j) function based on (20) 6: end for 7: Rearrange cost(j) in ascending order and store corresponding indexes in array cost_ordered 8: nodes = 1 9: while nodes ≤ M do 10: selected_nodes(nodes) = cost_ordered(nodes) 11: ρ (selected_nodes(nodes)) = 1 12: Calculate PD with selected SUs from selected_nodes array based on (6) 13: if PD ≥ β then 14: break 15: end if 16: if nodes = M then 17: ρ=0 18: break (‘‘PD is not ensured’’) 19: end if 20: nodes = nodes +1 21: end while 22: Compute ET based on energy objective functions in (21) and (22) 23: return ρ , ET Algorithm 1 is divided into two main sections: preprocessing and sensor selection phases. The first one, from line 1 to 7, determines statistical moments of distance d˜ 2 based on mobility model and size of simulation field. We also compute local detection and local false alarm probabilities and upper limit to sensor nodes M. In addition, cost function values on each SU are derived and rearranged from lowest to highest values associated with the priority to be awake. Total number of selected SUs represented by variable nodes is initialized to 1 as shown in line 8. Second phase of Algorithm 1 is composed by a w hile loop, from line 9 to 21, to compute which nodes will participate in CSS and which ones shall go to sleep mode to extend battery lifetime. This iterative cycle have to guarantee false alarm and detection global probabilities otherwise a feasible solution is not ensured. This loop shall return the solution given by vector ρ represented on variable selected_nodes. Global probability of detection PD is computed after iteratively including nodes with slowest cost function values. This inclusion ends when detection performance is achieved, line 13 to 15, or when total number of included nodes exceeds the upper limit M by testing this condition on lines 16 to 19. This iterative inclusion of nodes, line 10 and 11, will determine the selected ones to participate on CSS. Selected nodes are the ones to be activated by asserting the proper elements of vector ρ based on the obtained array selected_nodes by following rule in line 11. On each loop iteration, variable nodes is incremented by 1 as stated in line 20 until it reaches the upper limit M. Final step in the algorithm is to return awake SUs specified in the variable ρ ̸ = 0 and appropriate upper bound for consumed energy stored in the variable ET when a solution is reached.

7

6. Numerical results and discussion Extensive simulation results are presented in this section to validate our theoretical analysis over heterogeneous and homogeneous networks. We obtain the total number of awake SUs participating in CSS and consumed energy for each criterion given in Sections 4.1 and 4.2. Minor values of both parameters, total number of awake SUs and the corresponding consumed energy, are desired outputs to extend battery lifetime. Both parameters are determined for several simulation scenarios by varying network size, position of FC and PU, and total number of available SUs. Simulation scenarios are focused on illustrate performance based on total number of awake SUs and consumed energy by varying quantities above on Algorithm 1 for each specific case. Energy consumed is determined by evaluating (21) and (22). Initially, we assume that SUs are uniformly placed in a square field of side length s where the FC is located at the center of the field. After that, each SU follows the proposed mobility model based on Markov chains. Our √simulation field is inscribed in a circular cluster of radius Rc = 22 s to guarantee that each SU has equal SNR parameter value. The PU is located outside the cluster satisfying inequality shown in (26). For simplicity, the used freespace propagation model comprises isotropic antennas, for which GT = 1, GR = 1 and fc = 2.4 GHz. In addition, detection threshold for energy detector ϵ is selected as 2.5 times the noise variance σu2j , and global detection constraints are specified as α = 0.1 and β = 0.9 by following specifications in [4]. Energy consumption values in sensing and signal processing phases Esj are derived based on several models of Chipcon transceivers such as CC2400, CC2420, CC2430 and CC2500 [30], which satisfy IEEE 802.15.4 technical standard. Energy parameter Esj of cost function in (20) is commonly determined by adding two terms: energy to power the receiving electronic and energy related to signal processing operations. A typical value of 40 nJ is used to power the receiving electronic, signal processing phase consumes 122 nJ, 147 nJ, 200 nJ or 156 nJ depending on appropriate transceiver provided a data rate of 250 kB/s. Remaining energy parameters of network are defined by Et-elec = 80 nJ and eamp = 40.4 nJ/m2 similar to [31]. To illustrate performance, we compare active nodes and consumed energy of the proposed algorithm for heterogeneous networks from ‘‘wait-and-see" approach based on EESS static solution in [4]. Homogeneous networks will present similar results except that nodes to be activated may be chosen arbitrary. To compare EESS and proposed solutions based on statistical moments of consumed energy, results obtained from EESS are averaged for 10 000 simulation steps. On each figure, expected value criterion is represented by solid lines and dotted lines describe mixed criterion, while dashed lines style is used for ‘‘wait-and-see" approach based on EESS static algorithm. Fig. 3 depicts average total number of awake sensors to perform spectrum sensing and total energy consumption for varying network size ranged from 50 m to 300 m. Provided the variation of network size, then statistical moments of d˜ 2 will change appropriately, which in turn modify the statistical moments of total consumed energy E˜ T according to (21) and (22). Additionally, this simulation is performed by three different cases regarding location of FC to validate overall performance and adaptability of our solution. Position of FC is located at coordinate system origin (0, 0), at the center of simulation field (s/2, s/2) and at upper right vertex (s, s). Moreover, we simulate results with 100 SUs. Based on Fig. 3(a), ‘‘wait-and-see" approach tends to decrease the total number of active sensors as the network size is incremented when PU is located at a fixed position. Then, probability of detecting the PU increases when network size is larger owing to the fact that SUs have an increased probability to be closer to

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Fig. 3. Proposed algorithm for particular positions of FC. (a) Total number of active SUs. (b) Energy consumed in spectrum sensing phase.

PU. On the other hand, proposed criteria exhibit constant sensing nodes curves regardless a given FC location or network size due to a constant probability of detection is assumed for each SU. Proposed method establishes an upper bound to the total number of awake SUs for each FC location. This result indicates that proposed solution overestimates the total number of active SUs. Fig. 3(b) illustrates consumed energy in CSS of proposed solution and average values obtained by ‘‘wait-and-see" approach for varying network size and FC location. The larger the size of simulation field, then the bigger the value of consumed energy on proposed algorithm compared to average EESS solution. The increasing slope of proposed solution is due to the increase of network size. The larger the value of the network size, then the bigger the values of statistical moments of distance from each SU to FC. Expected value standard deviation criterion in (18) describes maximum energy consumption values of E˜ T . That means, by using this criterion, the corresponding curve in Fig. 3(b) sets an upper bound regarding random values of consumed energy. This results in a conservative criterion concerning moments of energy for extreme position of FC at (0, 0) and (s, s). However, when FC is located at the center of the simulation area, proposed solutions

Fig. 4. Simulations for varying positions of PU. (a) Total number of active SUs. (b) Energy consumed in spectrum sensing phase.

show similar behavior of consumed energy in spectrum sensing strategy compared to average EESS static solution. Fig. 4 exposes behavior of total number of awake SUs and energy consumption considering varying positions of PU, which in turn involves varying SNR parameter values on local probability of detection in (4). This scenario considers a network side of 200 m and 100 participating SUs. We have discarded all simulation scenarios where distance from PU to FC does not guarantee the requirements expressed in (26) related to equal SNR parameter for all SUs. In this cases, particular PU positions have significant influence on total number of selected nodes for both algorithms as shown in Fig. 4(a). Proposed solution returns more active nodes owing to assumptions on probabilities of detecting the PU signal. This result illustrates that proposed solution describes an upper bound to current behavior of active nodes. In a similar way, energy consumption levels of proposed solution tends to take increased distance from average EESS solution as the distance from PU to FC is augmented as depicted in Fig. 4(b). Expected value standard deviation criterion establishes the maximum values of consumed energy for each PU distance to FC. Fig. 5 represents active nodes and energy consumption levels considering different total number of participating SUs ranged

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9

Proposed criteria present higher values of awake nodes than ‘‘wait-and-see" approach based on EESS solution, however its simplicity allows to reduce processing time and data rate related to spectrum sensing decisions and the information results transmitted to FC provided that solution is computed only once. On the contrary, ‘‘wait-and-see" approach needs to be continuously applied on each time slot to have specifics solutions for each particular node positions. This results on a rather computationally expensive solution which the corresponding lacks applicability on real scenarios. 7. Conclusions The purpose of this paper was to address dynamic behavior of nodes on CRSN applications. An approach to model movement of sensors is described by Markov chains. Main contribution is the modeling of a stochastic optimization problem over dynamic CRSN. Consequently, two criteria were used to deal with the appropriate sensor selection in spectrum sensing: Expected Value and Expected Value Standard Deviation. These solutions represent a first approach to mobile sensor networks in order to minimize the total number of awake SUs participating in spectrum sensing to detect spectrum holes. Based on limited resources regarding computational and battery power capabilities of nodes, proposed stochastic approach offers solutions by means of minimizing total random energy consumption E˜ T through the analysis of its statistical moments. Simulation results validate performance of the proposed solution over several scenarios and its ability to properly describe tendency and upper bounds to total number of sensors and consumed energy for CSS. To consider additional and more realistic movement models, as well as their influence on this research, represents an open issue. Future work will be focused on the implementation of evolutionary optimization on dynamic environments that provides powerful heuristics rules to circumvent the uncertainty of node positions present in the real world. Fig. 5. Performance for varying number of participating SUs. (a) Total number of active SUs. (b) Energy consumed in spectrum sensing phase.

from 25 to 200. Proposed algorithm gives a constant number of active SUs despite the total number of participating nodes and network size as illustrated in Fig. 5(a). This can be explained by the fact that mobility models based on Markov chains establish the same opportunity for all SUs to occupy all possible locations over the simulation field. On the other hand, ‘‘waitand-see" approach provides the optimal solution to determine total number of active nodes for each time slot. This offers less awake nodes selected for CSS in ‘‘wait-and-see" approach as the total number of participating SUs is increased. The larger the total number of participating SUs, then the likely the proximity of SUs to FC and PU for a given network size, then the less the total number of active sensing nodes. Fig. 5(b) shows that proposed solutions exceed the average energy values obtained by EESS solution as network size is increased. In a similar way to previous figures, expected value and mixed criteria establish an upper bound to energy consumption in CSS. Energy consumption for ‘‘wait-and-see" approach has slightly lower values as the number of participating SUs increases due to a trade-off between awake nodes and distance from SUs to FC. Based on obtained results, proposed solution describes tendency and upper limits regarding total number of nodes and consumed energy in comparison with ‘‘wait-and-see" approach. Provided that upper limit fulfills requirements of problem formulation then proposed method represents a feasible solution.

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work is funded by the CONICYT, Chile-PFCHA/Doctorado Nacional/2016-21160292 and DICYT, Chile Project, Code 061813KC, Direction of Research, Development and Innovation, University of Santiago de Chile, USACH. References [1] M. Ibnkahla, Wireless Sensor Networks: A Cognitive Perspective, CRC Press, 2016. [2] X. Liu, M. Jia, Z. Na, W. Lu, F. Li, Multi-modal cooperative spectrum sensing based on dempster-shafer fusion in 5g-based cognitive radio, IEEE Access 6 (2018) 199–208, http://dx.doi.org/10.1109/ACCESS.2017.2761910. [3] S. Maleki, S.P. Chepuri, G. Leus, Optimization of hard fusion based spectrum sensing for energy-constrained cognitive radio networks, Phys. Commun. 9 (2013) 193–198, http://dx.doi.org/10.1016/j.phycom.2012.07.003. [4] M. Najimi, A. Ebrahimzadeh, S.M.H. Andargoli, A. Fallahi, A novel sensing nodes and decision node selection method for energy efficiency of cooperative spectrum sensing in cognitive sensor networks, IEEE Sens. J. 13 (5) (2013) 1610–1621, http://dx.doi.org/10.1109/JSEN.2013.2240900. [5] M. Najimi, A. Ebrahimzadeh, S.M.H. Andargoli, A. Fallahi, Lifetime maximization in cognitive sensor networks based on the node selection, IEEE Sens. J. 14 (7) (2014) 2376–2383, http://dx.doi.org/10.1109/JSEN.2014. 2311154.

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Karel Toledo de la Garza, Ph.D. candidate in Engineering Sciences, mention in Automation from the University of Santiago de Chile. He received the B.Sc. in Telecommunication Engineering and M.Sc. in Digital Systems degrees from the Technological University of Havana, Cuba, in 2012 and 2015, respectively. He is currently a research assistant in Wireless Sensor Networks (WSN) and Wireless Body Area Networks (WBAN) at University of Santiago de Chile. He has been with the Department of Telecommunications and Telematics, Technological University of Havana, CUJAE, Cuba, as a professor, from 2013 to 2015. His research interests include Cognitive Radio Sensor Networks (CRSN), energy efficient spectrum management, optimization and signal processing. Hector Kaschel Cárcamo, Dr.-Eng. in Electrical Engineering, University of Paderborn, Germany; Electrical Civil Engineer of the University of Santiago de Chile. Senior Member of the IEEE. He is currently a Professor of the Electrical Engineering Department at the University of Santiago de Chile. He has published more than 130 papers in national and international congresses and journals. His research interests include Industrial Communications Networks, Wireless Sensor Networks (WSN), Wireless Area Networks (WBAN), Smartcity, Smartgrid, Wireless Local Area Networks (WLAN) and Mobile Networks. Jorge Torres Gómez received the B.Sc., M.Sc., and Ph.D. degrees from the Technological University of Havana, CUJAE, Cuba, in 2008, 2010, and 2015, respectively. He is currently with the Department of Digital Signal Processing, Chemnitz University of Technology, Chemnitz, Germany. Since 2008, he has been with the School of Telecommunications and Electronics, CUJAE University, where he was a Lecturer with CUJAE, from 2008 to 2018. He is member of the Cuban Association of Pattern Recognition (ACRP). He has been with the Department of Signal Theory and Communications, Carlos III University of Madrid, Leganés Campus, Madrid, Spain, as guest lecturer. His research interests include digital signal processing, software defined radio, and wireless and wired communication systems.